Randomized iterative algorithms for solving a factorized linear system, $\mathbf A\mathbf B\mathbf x=\mathbf b$ with $\mathbf A\in{\mathbb{R}}^{m\times \ell}$, $\mathbf B\in{\mathbb{R}}^{\ell\times n}$, and $\mathbf b\in{\mathbb{R}}^m$, have recently been proposed. They take advantage of the factorized form and avoid forming the matrix $\mathbf C=\mathbf A\mathbf B$ explicitly. However, they can only find the minimum norm (least squares) solution. In contrast, the regularized randomized Kaczmarz (RRK) algorithm can find solutions with certain structures from consistent linear systems. In this work, by combining the randomized Kaczmarz algorithm or the randomized Gauss--Seidel algorithm with the RRK algorithm, we propose two novel regularized randomized iterative algorithms to find (least squares) solutions with certain structures of $\mathbf A\mathbf B\mathbf x=\mathbf b$. We prove linear convergence of the new algorithms. Computed examples are given to illustrate that the new algorithms can find sparse (least squares) solutions of $\mathbf A\mathbf B\mathbf x=\mathbf b$ and can be better than the existing randomized iterative algorithms for the corresponding full linear system $\mathbf C\mathbf x=\mathbf b$ with $\mathbf C=\mathbf A\mathbf B$.

翻译：最近，针对求解分解线性系统 $\mathbf A\mathbf B\mathbf x=\mathbf b$（其中 $\mathbf A\in{\mathbb{R}}^{m\times \ell}$，$\mathbf B\in{\mathbb{R}}^{\ell\times n}$，$\mathbf b\in{\mathbb{R}}^m$）提出了随机迭代算法。这些算法利用分解形式，避免了显式形成矩阵 $\mathbf C=\mathbf A\mathbf B$。然而，它们只能求解最小范数（最小二乘）解。相比之下，正则化随机Kaczmarz（RRK）算法能够从相容线性系统中找到具有特定结构的解。本文通过将随机Kaczmarz算法或随机Gauss-Seidel算法与RRK算法相结合，提出了两种新颖的正则化随机迭代算法，用于求解 $\mathbf A\mathbf B\mathbf x=\mathbf b$ 的具有特定结构的（最小二乘）解。我们证明了新算法的线性收敛性。通过计算实例说明，新算法能够求解 $\mathbf A\mathbf B\mathbf x=\mathbf b$ 的稀疏（最小二乘）解，并且对于相应的完整线性系统 $\mathbf C\mathbf x=\mathbf b$（其中 $\mathbf C=\mathbf A\mathbf B$），其性能优于现有的随机迭代算法。